Normal and Self-dual Normal Bases from Factorization of c xq+1 + d xq - ax - b

The present paper is interested in a family of normal bases, considered by V. M. Sidel’nikov, with the property that all the elements in a basis can be obtained from one element by repeatedly applying to it a linear fractional function of the form φ(x) = (ax + b)/(cx + d), a, b, c, d ∈ Fq. Sidel’nikov proved that the cross products for such a basis {αi} are of the form αiαj = ei−jαi+ ej−iαj +γ, i 6= j, where ek, γ ∈ Fq. We will show that every such basis can be formed by the roots of an irreducible factor of F (x) = cx + dx − ax− b. We will construct: (a) a normal basis of Fqn over Fq with complexity at most 3n− 2 for each divisor n of q− 1 and for n = p where p is the characteristic of Fq; (b) a self-dual normal basis of Fqn over Fq for n = p and for each odd divisor n of q − 1 or q + 1. When n = p, the self-dual normal basis constructed of Fqp over Fq also has complexity at most 3p − 2. In all cases, we will give the irreducible polynomials and the multiplication tables explicitly. Abbreviated title: Normal Bases. 1991 Mathematics subject classification: 11T30, 11T06.